Randomized techniques for approximating matrix decompositions may be used when a dataset is too large to efficiently solve a direct matrix decomposition. For example, the dataset may have too many observations (i.e., rows) and/or variables (i.e., columns) to solve using traditional techniques due to the amount of time or memory required to compute a solution. Two illustrative examples are the singular value decomposition (SVD) and a matrix decomposition that is inherent in a principal components analysis (hereafter denoted the principal components decomposition (PCD)). Of course, a drawback to using a decomposition based on randomization is that the resulting matrix factors are an approximation to the true matrix factors.